I call a set PECULIAR, if its elements are uncountable, pairwise disjoint subsets of R (the real number system). As for example, the set {(0,1),(3,5),[8,9]\Q},where Q denotes the set of rationals, is peculiar. My first question may have a very trivial answer that I cannot see immediately,that does there exist an uncountable, peculiar set? Next, I define the STRENGTH of a peculiar set to be the union of all its elements (which are sets). My question is that what are the subsets of R (if any), that are the strength of some uncountable peculiar set? In case complete specification seems impossible or meaningless, some examples will do.

Since the OP asked about sets of reals, and since these cannot be amorphous (because no amorphous set admits a linear ordering), let me point out that it is consistent with ZF that there is a Dedekind finite (hence not countable, as in Joel's answer) set of reals that cannot be partitioned into uncountably many many infinite pieces. One model in which this happens is the basic Cohen model, as defined in Section 5.3 of Jech's book "The Axiom of Choice". The relevant Dedekind finite set is the set of Cohen reals explicitly added by the forcing (the set Jech calls $A$). The fact that it cannot be split into uncountably many uncountable sets follows from work of Halpern and Levy ("The Boolean prime ideal theorem does not imply the axiom of choice," Axiomatic Set Theory, Part I, Proc. Symp. Pure Math. XIII, Part I (1971) pp. 83134). 


The comments have answered the question in the context of ZFC, where the axiom of choice is available. But there is something interesting to say when the axiom of choice fails (and this answer was too long for a comment). Namely, it is consistent with ZF that not every uncountable set can be partitioned into uncountably many disjoint uncountable sets. Indeed, it is consistent with ZF that an uncountable set cannot even be partitioned into two disjoint infinite sets. This is because it is consistent with ZF that there is an amorphous set, an infinite set all of whose subsets are either finite or the complement of a finite set. It follows that amorphous sets cannot be countable, since if there were an enumeration we could produce bad subsets by taking every other element. Thus, amorphous sets are uncountable sets with no partition into two or more disjoint infinite sets. 

